Lindblad equation

From Wikipedia, the free encyclopedia

In quantum mechanics, the Lindblad equation or master equation in the Lindblad form is a general type of master equation describing non-unitary (dissipative) evolution of the density matrix $ρ$ but preserving the trace, and complete positivity of $ρ$ . It reads:

$\dot\rho=-{i\over\hbar}[H,\rho]-{1\over\hbar}\sum_{n,m}h_{n,m}\big(\rho L_m L_n+L_m L_n\rho-2L_n\rho L_m\Big)+\mathrm{h.c.}$

where $\ \rho$ is the density matrix, $\ H$ is the Hamiltonian part, $\ L_m$ are operators defined by the system to model as are the constants $\ h_{n,m}$ . If the $\ L$ terms are all zero, then this is the ordinary (closed system) master equation, which is the quantum analog of the Liouville equation in classical mechanics. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has $\ L_0=a$ , $\ L_1=a^{\dagger}$ , $\ h_{0,1}=-(\gamma/2)(\bar n+1)$ , $\ h_{1,0}=-(\gamma/2)\bar n$ with all others $\ h_{n,m}=0$ . Here $\bar n$ is the mean number of excitations in the reservoir damping the oscillator and $\ \gamma$ is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.